{
 "cells": [
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [],
   "source": [
    "import math\n",
    "import time\n",
    "import numpy as np\n",
    "import torch\n",
    "from d2l import torch as d2l"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [],
   "source": [
    "n = 100000\n",
    "a = torch.ones(n)\n",
    "b = torch.ones(n)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [],
   "source": [
    "class Timer:\n",
    "    def __init__(self):\n",
    "        self.times = []\n",
    "        self.start()\n",
    "        \n",
    "    def start(self):\n",
    "        \"\"\"启动计时器\"\"\"\n",
    "        self.tik = time.time()\n",
    "        \n",
    "    def stop(self):\n",
    "        \"\"\"停止计时器并将时间记录在列表中\"\"\"\n",
    "        self.times.append(time.time() - self.tik)\n",
    "        return self.times[-1]\n",
    "\n",
    "    def avg(self):\n",
    "        \"\"\"返回平均时间\"\"\"\n",
    "        return sum(self.times) / len(self.times)\n",
    "    \n",
    "    def sum(self):\n",
    "        \"\"\"返回时间总和\"\"\"\n",
    "        return sum(self.times)\n",
    "    \n",
    "    def cumsum(self):\n",
    "        \"\"\"返回累加时间\"\"\"\n",
    "        return np.array(self.times).cumsum().tolist()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "'0.77594995 sec'"
      ]
     },
     "execution_count": 4,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "c = torch.zeros(n)\n",
    "timer = Timer()\n",
    "for i in range(n):\n",
    "    c[i] = a[i] + b[i]\n",
    "f'{timer.stop():.8f} sec' "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "'0.00099564 sec'"
      ]
     },
     "execution_count": 5,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "timer.start()\n",
    "d = a + b\n",
    "f'{timer.stop():.8f} sec'"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [],
   "source": [
    "def normal(x, mu, sigma):\n",
    "    p = 1 / math.sqrt(2 * math.pi * sigma**2)\n",
    "    return p * np.exp(-0.5 / sigma**2 * (x - mu)**2)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [
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       "</svg>\r\n"
      ],
      "text/plain": [
       "<Figure size 324x180 with 1 Axes>"
      ]
     },
     "metadata": {
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     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "x = np.arange(-7, 12, 0.01)\n",
    "\n",
    "# 均值和标准差对\n",
    "params = [(0, 1), (1, 2), (3, 1),(4, 0.5)]\n",
    "d2l.plot(x, [normal(x, mu, sigma) for mu, sigma in params], xlabel='x',\n",
    "         ylabel='p(x)', figsize=(4.5, 2.5),\n",
    "         legend=[f'mean {mu}, std {sigma}' for mu, sigma in params])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# 练习"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## 第一题"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<font size=5>$min\\sum (x_{i}-b)^{2}$  \n",
    "    转为优化问题即求方程的最小值\n",
    "    $\\Rightarrow \\hat{y}=\\sum (x_{i}-b)^{2}\n",
    "\\Rightarrow \\partial x_{i}=0$  \n",
    "$\\partial x_{i}=2(\\sum x_{i}-nb)$  \n",
    "    $b=\\frac{\\sum x_{i}}{n}$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<font size=5>其损失函数为$l^{i}(b)=\\frac{1}{2}(\\hat{y}^{(i)}-y^{(i)})^2$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<font size=5>引入误差项$\\varepsilon$ 且 $\\varepsilon \\sim N(0,\\sigma ^2)$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$\\hat{y}=\\sum (x_{i}-b)^{2}+\\varepsilon$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "带入正态分布函数"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$P(\\textbf{y}\\mid \\textbf{X})=\\frac{1}{\\sqrt{2\\pi \\sigma ^2}}exp(-\\frac{(y-\\sum (x_{i}-b)^{2})^{2}}{2\\sigma ^2})$  \n",
    "两边取对数$-log$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$-logP(\\textbf{y}\\mid \\textbf{X})=\\sum_{i=1}^{n}\n",
    "\\frac{1}{2}log(2\\pi \\sigma ^2)+\\frac{1}{2\\sigma ^2}(y-\\sum (x_{i}-b)^{2})^{2}$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "假设$\\sigma$为常数，则上式最终的结果与损失函数一致。"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## 第二题"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# 线性回归从0实现"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [],
   "source": [
    "import random\n",
    "import torch\n",
    "from d2l import torch as d2l"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [],
   "source": [
    "def synthetic_data(w, b, num_examples):\n",
    "    \n",
    "    X = torch.normal(0, 1, (num_examples, len(w)))\n",
    "    y = torch.matmul(X, w) + b\n",
    "    y += torch.normal(0, 0.01, y.shape)\n",
    "    return X, y.reshape((-1, 1))\n",
    "\n",
    "true_w = torch.tensor([2, -3.4])\n",
    "true_b = 4.2\n",
    "features, labels = synthetic_data(true_w, true_b, 1000)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "features: tensor([0.4531, 1.1236]) \n",
      "label: tensor([1.2844])\n"
     ]
    }
   ],
   "source": [
    "print('features:', features[0], '\\nlabel:', labels[0])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [
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   "source": [
    "d2l.set_figsize()\n",
    "d2l.plt.scatter(features[:, 1].detach().numpy(),\n",
    "                labels.detach().numpy(), 1);"
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   "execution_count": 8,
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    "def data_iter(batch_size, features, labels):\n",
    "    num_examples = len(features)\n",
    "    indices = list(range(num_examples))\n",
    "    # 这些样本是随机读取的，没有特定的顺序\n",
    "    random.shuffle(indices)\n",
    "    for i in range(0, num_examples, batch_size):\n",
    "        batch_indices = torch.tensor(indices[i:min(i + batch_size, num_examples)])\n",
    "        yield features[batch_indices], labels[batch_indices]"
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      "        [ 2.3442,  1.5682],\n",
      "        [ 0.3558, -0.7973],\n",
      "        [ 0.1043,  0.4337],\n",
      "        [ 0.4326, -0.2264],\n",
      "        [ 0.1185, -0.3915],\n",
      "        [ 1.1654,  0.0220],\n",
      "        [-0.4678,  1.1290],\n",
      "        [ 0.0893, -0.2794],\n",
      "        [ 1.4300, -0.4625],\n",
      "        [ 0.3145,  0.0229],\n",
      "        [-0.0129, -0.1741]]) \n",
      " tensor([[-1.5141],\n",
      "        [10.1309],\n",
      "        [ 3.5726],\n",
      "        [ 7.6088],\n",
      "        [ 2.9531],\n",
      "        [ 5.8443],\n",
      "        [ 5.7524],\n",
      "        [ 6.4447],\n",
      "        [-0.5540],\n",
      "        [ 5.3428],\n",
      "        [ 8.6287],\n",
      "        [ 4.7455],\n",
      "        [ 4.7526]])\n",
      "tensor([[-1.7722,  1.1666],\n",
      "        [ 0.0313,  1.2894],\n",
      "        [ 0.4537, -0.7051],\n",
      "        [-0.9979, -0.6779],\n",
      "        [ 1.9228,  0.9647],\n",
      "        [-1.1560, -1.1150],\n",
      "        [-0.3119,  2.8957],\n",
      "        [-0.0772,  0.6839],\n",
      "        [-2.3065, -0.3241],\n",
      "        [-1.4183, -2.1921],\n",
      "        [-1.8308, -1.5215],\n",
      "        [-0.1676, -1.4569],\n",
      "        [ 0.0444,  0.5840]]) \n",
      " tensor([[-3.3099],\n",
      "        [-0.1290],\n",
      "        [ 7.5201],\n",
      "        [ 4.5070],\n",
      "        [ 4.7677],\n",
      "        [ 5.6827],\n",
      "        [-6.2818],\n",
      "        [ 1.7083],\n",
      "        [ 0.6872],\n",
      "        [ 8.8046],\n",
      "        [ 5.7079],\n",
      "        [ 8.8285],\n",
      "        [ 2.3082]])\n",
      "tensor([[-0.4969,  0.0913],\n",
      "        [-1.4627, -1.9791],\n",
      "        [ 1.2880, -0.1223],\n",
      "        [-0.1283,  1.1633],\n",
      "        [-0.1127,  1.4054],\n",
      "        [-0.5441, -0.1146],\n",
      "        [ 0.8337,  0.9537],\n",
      "        [ 0.1310, -0.9502],\n",
      "        [-0.2217,  1.3895],\n",
      "        [-0.6295, -0.6652],\n",
      "        [ 0.2584, -0.5624],\n",
      "        [ 1.3268, -0.1790],\n",
      "        [ 1.1303, -0.4545]]) \n",
      " tensor([[ 2.8776],\n",
      "        [ 7.9921],\n",
      "        [ 7.1718],\n",
      "        [-0.0234],\n",
      "        [-0.8107],\n",
      "        [ 3.4971],\n",
      "        [ 2.6166],\n",
      "        [ 7.6834],\n",
      "        [-0.9712],\n",
      "        [ 5.2159],\n",
      "        [ 6.6233],\n",
      "        [ 7.4633],\n",
      "        [ 8.0036]])\n",
      "tensor([[ 0.0841, -0.2539],\n",
      "        [ 0.8030,  0.6436],\n",
      "        [ 0.3717, -0.7296],\n",
      "        [ 0.7219, -0.6003],\n",
      "        [-1.0891,  0.4734],\n",
      "        [ 0.6146,  0.1926],\n",
      "        [-1.4954, -0.6912],\n",
      "        [-0.0868, -0.4898],\n",
      "        [-0.5283, -0.6069],\n",
      "        [-0.9962, -0.5183],\n",
      "        [ 0.0890, -0.0309],\n",
      "        [ 0.9930,  1.4164],\n",
      "        [ 0.0647, -0.4518]]) \n",
      " tensor([[5.2405],\n",
      "        [3.6062],\n",
      "        [7.4253],\n",
      "        [7.6972],\n",
      "        [0.4054],\n",
      "        [4.7872],\n",
      "        [3.5611],\n",
      "        [5.7123],\n",
      "        [5.2138],\n",
      "        [3.9664],\n",
      "        [4.4947],\n",
      "        [1.3826],\n",
      "        [5.8680]])\n",
      "tensor([[ 1.1941, -1.6481],\n",
      "        [ 0.3562, -0.5032],\n",
      "        [ 1.1822, -1.6047],\n",
      "        [-1.1954,  2.2317],\n",
      "        [ 1.7763, -0.4246],\n",
      "        [ 0.7746, -1.1495],\n",
      "        [-0.1021, -0.3260],\n",
      "        [ 0.4047, -2.0439],\n",
      "        [ 1.3641,  0.2561],\n",
      "        [-0.8875,  1.1063],\n",
      "        [ 1.3723,  1.0358],\n",
      "        [-0.9630,  1.4952],\n",
      "        [-2.4955, -0.6553]]) \n",
      " tensor([[12.1888],\n",
      "        [ 6.6193],\n",
      "        [12.0247],\n",
      "        [-5.7865],\n",
      "        [ 9.2010],\n",
      "        [ 9.6655],\n",
      "        [ 5.1031],\n",
      "        [11.9468],\n",
      "        [ 6.0461],\n",
      "        [-1.3361],\n",
      "        [ 3.4055],\n",
      "        [-2.8163],\n",
      "        [ 1.4425]])\n",
      "tensor([[-0.4146, -1.4327],\n",
      "        [ 0.4832,  1.3962],\n",
      "        [-0.2683,  0.6638],\n",
      "        [ 0.5960, -0.8713],\n",
      "        [-0.0887, -0.4267],\n",
      "        [ 0.9362, -0.7036],\n",
      "        [-0.4844, -0.1317],\n",
      "        [ 1.8986,  0.6573],\n",
      "        [ 0.1537,  0.2151],\n",
      "        [ 1.0519,  0.2490],\n",
      "        [-0.6957,  1.2660],\n",
      "        [ 0.5848, -1.3583],\n",
      "        [ 1.3858, -0.3178]]) \n",
      " tensor([[ 8.2359],\n",
      "        [ 0.4153],\n",
      "        [ 1.4225],\n",
      "        [ 8.3440],\n",
      "        [ 5.4763],\n",
      "        [ 8.4594],\n",
      "        [ 3.6702],\n",
      "        [ 5.7648],\n",
      "        [ 3.7871],\n",
      "        [ 5.4653],\n",
      "        [-1.4886],\n",
      "        [ 9.9825],\n",
      "        [ 8.0537]])\n",
      "tensor([[ 0.0175,  0.4924],\n",
      "        [-0.1652,  0.5659],\n",
      "        [ 0.5028,  0.5057],\n",
      "        [-0.3644,  0.4091],\n",
      "        [-1.0165, -0.8508],\n",
      "        [-0.5144,  0.4977],\n",
      "        [ 1.4557,  1.8226],\n",
      "        [-0.6073,  0.1679],\n",
      "        [ 2.3070,  0.9516],\n",
      "        [-0.4402, -0.2831],\n",
      "        [ 0.9034, -1.3724],\n",
      "        [-0.3416,  0.5139],\n",
      "        [ 1.0126, -0.4829]]) \n",
      " tensor([[ 2.5562],\n",
      "        [ 1.9615],\n",
      "        [ 3.4792],\n",
      "        [ 2.0769],\n",
      "        [ 5.0688],\n",
      "        [ 1.5106],\n",
      "        [ 0.9138],\n",
      "        [ 2.4112],\n",
      "        [ 5.5901],\n",
      "        [ 4.2665],\n",
      "        [10.6875],\n",
      "        [ 1.7734],\n",
      "        [ 7.8538]])\n",
      "tensor([[-1.0931,  0.9529],\n",
      "        [-1.2571, -1.6933],\n",
      "        [ 0.3923, -0.2220],\n",
      "        [ 0.7742,  0.9660],\n",
      "        [ 1.2047, -1.5749],\n",
      "        [ 0.6097,  0.5416],\n",
      "        [-0.1532, -1.4375],\n",
      "        [ 0.0084, -1.2705],\n",
      "        [-1.8528, -1.3254],\n",
      "        [ 1.2808,  0.7092],\n",
      "        [ 0.2255,  0.8015],\n",
      "        [-0.3402,  1.9362],\n",
      "        [-0.6242,  0.4103]]) \n",
      " tensor([[-1.2362],\n",
      "        [ 7.4445],\n",
      "        [ 5.7526],\n",
      "        [ 2.4576],\n",
      "        [11.9528],\n",
      "        [ 3.5768],\n",
      "        [ 8.7990],\n",
      "        [ 8.5253],\n",
      "        [ 5.0110],\n",
      "        [ 4.3525],\n",
      "        [ 1.9267],\n",
      "        [-3.0550],\n",
      "        [ 1.5702]])\n",
      "tensor([[ 0.0400, -0.5768],\n",
      "        [ 1.5043, -0.0713],\n",
      "        [-0.5125, -0.5568],\n",
      "        [-1.6033, -0.0343],\n",
      "        [-0.2805, -1.1709],\n",
      "        [ 0.3514, -0.5213],\n",
      "        [-1.0803, -0.8337],\n",
      "        [ 0.3040, -1.5622],\n",
      "        [-1.0803,  0.8991],\n",
      "        [ 2.1626,  1.0770],\n",
      "        [ 0.1675,  0.9936],\n",
      "        [-0.9722,  1.6581],\n",
      "        [ 0.7200, -1.1143]]) \n",
      " tensor([[ 6.2445],\n",
      "        [ 7.4478],\n",
      "        [ 5.0665],\n",
      "        [ 1.0855],\n",
      "        [ 7.6125],\n",
      "        [ 6.6939],\n",
      "        [ 4.8735],\n",
      "        [10.1105],\n",
      "        [-1.0218],\n",
      "        [ 4.8630],\n",
      "        [ 1.1580],\n",
      "        [-3.3800],\n",
      "        [ 9.4134]])\n",
      "tensor([[ 1.0366, -0.0529],\n",
      "        [-1.4591, -0.3139],\n",
      "        [ 0.6856,  2.0625],\n",
      "        [-0.8728, -0.5553],\n",
      "        [-0.0135, -0.1514],\n",
      "        [-0.4594,  0.0669],\n",
      "        [-0.9143, -0.3099],\n",
      "        [-0.5422,  1.5433],\n",
      "        [-0.9692, -0.2482],\n",
      "        [ 0.5518, -0.5620],\n",
      "        [ 1.3640,  1.6014],\n",
      "        [ 0.3762,  1.5160],\n",
      "        [ 0.6288, -1.1745]]) \n",
      " tensor([[ 6.4445],\n",
      "        [ 2.3435],\n",
      "        [-1.4293],\n",
      "        [ 4.3410],\n",
      "        [ 4.6915],\n",
      "        [ 3.0401],\n",
      "        [ 3.4172],\n",
      "        [-2.1418],\n",
      "        [ 3.0994],\n",
      "        [ 7.2015],\n",
      "        [ 1.4743],\n",
      "        [-0.2186],\n",
      "        [ 9.4549]])\n",
      "tensor([[ 0.5484,  0.5616],\n",
      "        [ 0.1953,  0.3834],\n",
      "        [ 0.0430,  0.1815],\n",
      "        [ 1.4223, -0.1695],\n",
      "        [-1.1018, -0.2673],\n",
      "        [ 1.6207, -1.1441],\n",
      "        [ 1.0929,  0.9482],\n",
      "        [-2.0096,  1.4588],\n",
      "        [ 0.7667,  0.1540],\n",
      "        [ 0.2277,  0.7977],\n",
      "        [ 0.4553, -0.9431],\n",
      "        [-1.2716, -0.4286],\n",
      "        [-0.7528, -1.8862]]) \n",
      " tensor([[ 3.3790],\n",
      "        [ 3.2875],\n",
      "        [ 3.6588],\n",
      "        [ 7.6176],\n",
      "        [ 2.9181],\n",
      "        [11.3291],\n",
      "        [ 3.1573],\n",
      "        [-4.7804],\n",
      "        [ 5.2070],\n",
      "        [ 1.9345],\n",
      "        [ 8.3169],\n",
      "        [ 3.0935],\n",
      "        [ 9.1111]])\n",
      "tensor([[ 0.9884, -0.7677],\n",
      "        [ 0.8986,  0.0687],\n",
      "        [ 2.8957, -0.9582],\n",
      "        [ 0.3669, -0.0334],\n",
      "        [ 0.8337,  1.8464],\n",
      "        [ 0.6326,  0.1863],\n",
      "        [ 1.3503,  0.5777],\n",
      "        [-1.5872,  2.6747],\n",
      "        [ 0.5415, -0.0985],\n",
      "        [-0.4502,  0.8291],\n",
      "        [ 0.2487, -0.2092],\n",
      "        [-2.1450, -1.7577]]) \n",
      " tensor([[ 8.7922],\n",
      "        [ 5.7649],\n",
      "        [13.2587],\n",
      "        [ 5.0473],\n",
      "        [-0.4076],\n",
      "        [ 4.8311],\n",
      "        [ 4.9403],\n",
      "        [-8.0733],\n",
      "        [ 5.6309],\n",
      "        [ 0.4832],\n",
      "        [ 5.4084],\n",
      "        [ 5.8817]])\n"
     ]
    }
   ],
   "source": [
    "batch_size = 10\n",
    "for X, y in data_iter(batch_size, features, labels):\n",
    "    print(X, '\\n', y)\n",
    "    "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "初始化模型参数"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {},
   "outputs": [],
   "source": [
    "# w = torch.zeros([2, 1], dtype=torch.float32)\n",
    "w = torch.normal(0, 0.01, size=(2, 1), requires_grad=True)\n",
    "b = torch.zeros(1, requires_grad=True)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "定义模型"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {},
   "outputs": [],
   "source": [
    "def linreg(X, w, b):\n",
    "    \"\"\"线性回归模型\"\"\"\n",
    "    return torch.matmul(X, w) + b"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "定义损失函数"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {},
   "outputs": [],
   "source": [
    "def squared_loss(y_hat, y):\n",
    "    \"\"\"均方根损失函数\"\"\"\n",
    "    return (y_hat - y.reshape(y_hat.shape))**2 / 2"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "定义优化算法"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {},
   "outputs": [],
   "source": [
    "def sgd(params, lr, batch_size):\n",
    "    \n",
    "    with torch.no_grad():\n",
    "        for param in params:\n",
    "            param -= lr * param.grad / batch_size\n",
    "            param.grad.zero_()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "epoch 1, loss 0.000050\n",
      "epoch 2, loss 0.000031\n",
      "epoch 3, loss 0.000034\n"
     ]
    }
   ],
   "source": [
    "lr = 0.1\n",
    "net = linreg\n",
    "num_epochs = 3\n",
    "loss = squared_loss\n",
    "\n",
    "for epoch in range(num_epochs):\n",
    "    for X, y in data_iter(batch_size, features, labels):\n",
    "        l = loss(net(X, w, b), y)\n",
    "        l.sum().backward()\n",
    "        sgd([w, b], lr, batch_size)\n",
    "    with torch.no_grad():\n",
    "        train_loss = loss(net(X, w, b), y)\n",
    "        print(f'epoch {epoch + 1}, loss {float(train_loss.mean()):f}')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 25,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "w的估计误差：tensor([ 0.0005, -0.0011], grad_fn=<SubBackward0>)\n",
      "b的估计误差：tensor([-0.0006], grad_fn=<RsubBackward1>)\n"
     ]
    }
   ],
   "source": [
    "print(f'w的估计误差：{(true_w - w.reshape(true_w.shape))}')\n",
    "print(f'b的估计误差：{(true_b - b)}')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 42,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "w tensor([[0.],\n",
      "        [0.]])\n"
     ]
    }
   ],
   "source": [
    "print(f'w {w}')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 43,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "b tensor([0.], requires_grad=True)\n"
     ]
    }
   ],
   "source": [
    "print(f'b {b}')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# 练习"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "第一题，如果我们将权重初始化为零，会发生什么。算法仍然有效吗？"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "不会。"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "第二题，假设你是 乔治·西蒙·欧姆 ，试图为电压和电流的关系建立一个模型。你能使用自动微分来学习模型的参数吗?"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "def ohm(U, w, b):\n",
    "    return U / w - b"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "第三题，您能基于 普朗克定律 使用光谱能量密度来确定物体的温度吗？"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$$\n",
    "B(\\lambda, T)=\\frac{2 h c^{2}}{\\lambda^{5}} \\frac{1}{e^{\\frac{h c}{\\lambda k_{\\mathrm{B}} T}}-1}  \n",
    "$$ 普朗克定律"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$$\n",
    "u_{i}(T)=\\frac{4 \\pi}{c} B_{i}(T)\n",
    "$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "式中，$u_{i}(T)$为光谱能量密度，$B_{i}(T)$为总辐射度关于温度（T）的函数（普朗克定律），$i\\in \\lambda $，$h$是普朗克常熟，$c$为光速，$k$为玻尔兹曼常数"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "合并常数项，因此上式在第i波段的光谱能量密度可转为\n",
    "$$\n",
    "u_{i}(T)=\\frac{6 \\pi h c}{i^{5}} \\frac{1}{e^{\\frac{h c}{i k_{\\mathrm{B}} T}}-1} = l_{1} \\frac{1}{e^{\\frac{l_{2}}{T}}-1}\n",
    "$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$$\n",
    "l_{1}=\\frac{6 \\pi h c}{i^{5}}\n",
    "$$  \n",
    "$$\n",
    "l_{2}=\\frac{h c}{i k_{\\mathrm{B}}}\n",
    "$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$$\n",
    "T=\\frac{l_{2}}{ln(\\frac{l_{1}}{u})+1}\n",
    "$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "第四题，如果你想计算二阶导数可能会遇到什么问题？你会如何解决这些问题？"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": []
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "第五题，为什么在 squared_loss 函数中需要使用 reshape 函数？"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "答：y可能是向量，而y_hat可能是多行一列的矩阵"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "第七题，如果样本个数不能被批量大小整除，data_iter函数的行为会有什么变化？"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "答：test中，data_iter最后会取剩余的个数来计算结果"
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